3. The peak spectra
A large fraction of AGN is variable at -ray energies. Any cut-off arising from opacity effects will be more prominent at high flux levels since then the intrinsic photon density of the source is high. We have therefore chosen a subsample of AGN for which at least a moderate level of variability can be found. These sources are indicated in the variability column in Table 1. The light curves of the individual sources have been taken from the second EGRET catalogue (Thompson et al., 1995) and from the standard viewing period analysis of the EGRET group at MPE. In total we are left with 28 FSRQs and only 6 BL Lacs which are variable. The analysis is now similar to that described in the previous section except that the spectra are not derived on the basis of the summed data of Phases I-III but only on data of the viewing periods in which the sources showed the highest flux levels. We should note that detections with formal significance have not been considered. In case of two or more viewing periods of comparable duration and comparable source flux preference has been given to data taken with smaller aspect angle. The results for the summed spectra are presented in Fig.4 while the summed power-law fit statistic is shown in Fig.5.
There is not very much change compared to the average behaviour in case of BL Lacs. One has to keep in mind that we are now left with 6 objects and the statistics are not sufficient to distinguish general trends from pathological individuals.
The behaviour of FSRQs during their peak phase is more interesting. At first we see that the flare spectra are harder than the average, at least between 100 MeV and a few GeV. This is a confirmation of a claim by Mücke et al. (1996) who found a hardening of the -ray spectra with increasing flux level for 8 highly variable AGN. When fitting a power-law to the intensity ratio of average FSRQ at peak and total average FSRQ we find the spectrum of the average FSRQ at peak harder than that of the time-average by when all energy bins are considered and by when only the data with very good statistics between 70 MeV and 4 GeV are considered. The goodness-of-fit for a constant is and , respectively, which corresponds to about 2.7 significance. A Fischer-Snedecor-test indicates that with 4.0 significance a linear relation is a better fit to the intensity ratio than a constant.
We also see that at energies below 70 MeV and at energies above 4 GeV the peak spectra show some evidence of a cut-off which is more prominent in summed fit statistic than in the summed intensity. It should be noted that the error bars for the summed intensity are mainly determined by sources with poor statistics, i.e. short exposures or large off-axis angles. These sources usually contribute very little to the summed fit statistic since both at low and high energies the uncertainty in counts exceeds the number of detected counts. Thus the that goes into the summation in Eq.1 is practically restricted to . This is different for sources with good statistics for which even at highest energies the number of observed and expected counts is roughly between 0.5 and 10. Thus the summed fit statistic is dominated by sources with good statistics and its results can not be compared directly to the summed intensity spectra.
At high energies a cut-off would be the expected behaviour since the strength of opacity effects depends directly on the flux level. Though the low-energy deficit is significant at 50-70 MeV the average flux per source at flare state is still a factor of 2.0 0.6 higher than in the time-averaged case (cf. Fig.1). The flux per source at 30-50 MeV is the same as in the time-averaged case. Thus it appears that the -ray flux below 50 MeV does not change, or at least not in phase with the -ray flux above 100 MeV.
If the -ray flaring spectrum of FSRQ is indeed a power-law above 70 MeV, which is then cut off at a few GeV due to increasing opacity, we may underestimate the significance level of this cut-off in our analysis. The reason is that the power-law fit tries to fit both the power-law part and the cut-offs, and therefore will underestimate the power-law part at medium energies which is in principle the null hypothesis in the statistical test for cut-offs. This effect can be observed in Fig.5 as a strong preponderance of positive deviations between 100 MeV and 4 GeV.
To get a better idea of the significance level of the cut-offs we have repeated the power-law fits for the peak phases of FSRQs under the constraint that now the fit is based on the energy band of 70 MeV to 4 GeV and then extended to calculate the true deviations in the outer energy bands. The result is shown in Fig.6. At energies below 70 MeV there is a deficiency of intensity compared to power-law behaviour with total statistical significance of 3.0 while at high energies above 4 GeV we observe an intensity deficit with 2.6 significance. As mentioned before the statistical significance has to be inferred from the summed fit statistic and it cannot be easily compared to the summed intensity spectrum.
This result is stable against the choice of source. The highest flux level of a source does not necessarily imply the highest significance. We have tested whether our analysis is influenced by data with poor statistics by excluding all sources with formal significance of less than 6 in the viewing period of peak flux. There was no change in the summed fit statistic beyond small statistical fluctuations. On the other hand, some sources have strong secondary maxima in their -ray light curves or have been observed at similar flux levels in adjacent viewing periods. To do the opposite test we have extended our data base by including all observations which yielded an integrated flux of or for which the derived flux was within 2 of the peak value. Again our conclusion remained unchanged. The problem with the latter test is that though the statistical basis improves, individual sources, which appear three or four times, start to influence the result. The fact that our result is indeed robust and does not depend on a specific selection criteria, gives us confidence that it is real.
3.1. Possible systematics
Pulsar data show that at low energies the effective area of EGRET needs to be decreased from the value determined by the pre-flight calibrations. The accuracy of this correction factor can be estimated to be better than 30% at 30 MeV to 50 MeV and better than 10% at 50 MeV to 70 MeV (Fierro 1995, 3.4). There is another source of uncertainty at low energies which has to do with spill-over. At -ray energies below 100 MeV EGRET's effective area decreases rapidly (Thompson et al. 1993). Hence photons with original energy of around 100 MeV, which are misidentified as of lower energy, get a strong weight and mimic a high intensity at low energies. For the typical spectrum around 10% of the low energy photons are due to this spill-over. Standard EGRET software for spectral fits accounts for this spill-over as far as the relation between counts and intensity is concerned. However this effect also influences the effective point-spread function at low energies which is calculated under the assumption of an spectrum. Since the possible error in the point-spread function is around 10% we expect corresponding uncertainties in the likelihood fits of similar order. In total we estimate less than 50% systematical uncertainties in the intensity level at lowest -ray energies and around 20% in the second energy band which is to be compared to the total statistical uncertainty in the summed intensity spectrum of quasars in Fig.4 where we have 130% at 30 MeV to 50 MeV and 32% at 50 MeV to 70 MeV. In both cases the statistical uncertainties are much larger than the systematic uncertainties so that the former are a fair measure of the total uncertainty at low -ray energies.
At high -ray energies the spill-over does not play a strong role, since EGRET's effective area changes only weakly with energy in this range. Nevertheless possible calibration errors of effective area and point-spread function may influence the result. Also if a significant fraction of the -ray sources is misidentified, then by using the positions of the radio sources we may underestimate the number of observed photons at high energies where the point-spread function is most narrow. Here it is instructive to compare Fig.6 to the corresponding result for the average emission of FSRQs in Fig.3. There we had no signal for a high-energy cut-off. We have tested this also for the case that the power-law fit is derived only between 70 MeV and 4 GeV (as in Fig.6) with similar result: there is not even a 1 indication for a cut-off beyond 4 GeV. Since any systematic effect should influence also the average FSRQ spectrum we can be sure that the observed high energy cut-off in the peak FSRQ spectrum is real. A misidentification of the -ray sources with FSRQ in general, such that the -ray spectra discussed here are not those of FSRQ, has been found extremely unlikely in statistical studies (Mattox et al. 1997).
We have tested the reliability of our method by Monte-Carlo simulations. These simulations would detect systematic problems in the analysis tools, which may arise from the small photon numbers both at low and at high -ray energies. We did not detect significant systematic deviations from a Gaussian distribution of the variable of Eq.1.
© European Southern Observatory (ESO) 1997
Online publication: April 20, 1998